How do you turn a repeating decimal into a fraction? - Answers

There are different methods. Algebraically as follows:ex: 0.272727.....write a simple equation x = 0.272727... notice 2 digits are repeating so multiply by 100 (1 with 2 zeros because 2 digits repeat)100 x = 27.272727.... write the original equation below this and subtractx = 0.272727...99x = 27 notice all the decimals cancel. now divide by the coefficient but don't really divide just write as a fraction99x/99 = 27/99 reduce the fractionx = 3/11I will give you an example that my son had in one of his tests and he found it difficult, but nothing is difficult if you apply your knowledge to solve it:13.0123454545...= 13.0123 + 0. 0000454545...= 13.0123 + 0.454545... x 1/10000 now try to write 0.454545... as a fractionlet x = 0.454545... (1) multiply by 100 both sides100x = 45.454545... (2) subtract (1) from (2)99x = 45 divide both sides by 99x = 45/99Thus, 0.454545... = 45/99 replace it into the expression above= 130123/10000 + 45/990000= [(99)(130123) + 45]/990000= (12882177 + 45)/990000= 12882222/990000--------------------------------------take the repeating decimal and count how many digits repeat.multiply it by a power of 10 that has the same number of zeros as repeating digits; eg if 2 digits repeat multiple by 100subtract the original from the multiplied version.put the result over the multiplier in step 2 less 1if there is a decimal point in the numerator, multiply both top and bottom by the same power of 10 to get rid of the decimal pointsimplify the fraction.Using the examples above: 0.272727.... has 2 digits repeating, so multiply by 1000.272727... × 100 = 27.2727...Subtract to get: 27.272727... - 0.272727... = 27Put over 100 -1 = 99: 27/99simplify: 27/99 = (9×3)/(9×11) = 3/110.1666... has 1 repeating digit so multiply by 100.1666 × 10 = 1.666...subtract to get 1.666... - 0.1666... = 1.5Put over 10 - 1 = 9: 1.6/9As the numerator contains 1 decimal place multiply top and bottom by 10 to get rid of the decimal in the numerator: 1.5/9 × 10/10 = 15/90Simplify: 15/90 = (15×1)/(15×6) = 1/713.0123454545... has 2 repeating digits, so multiply by 10013.0123454545... × 100 = 1301.234545...Subtract to get: 1301.234545... - 13.0123454545... = 1288.2222Put over 100 - 1 = 99: 1288.2222/99As numerator contains 4 decimal places, multiply top and bottom by 10000 to get rid of the decimal in the numerator: 1288.2222/99 × 10000/10000 = 12882222/990000Simplify: 12882222/990000 = (18×715679)/(18×55000) = 715679/55000 = 13 679/55000